**
Introduction to Advanced Mathematics, MATH 215, Fall 2017
**

**Instructor:** Daniel Groves, 727 SEO e.mail

** Course webpage:**
http://www.math.uic.edu/~groves/teaching/2017-18/215/

**Syllabus**: Download here.

**Course hours:**

MWF, 11:00-11:50AM, Room 310, Addams Hall.

** Office hours:**

Mondays 10am, Wednesdays 12pm.

(Or, you can make an apointment by e.mail or try stopping past my office.)

**Text:** "An Introduction to Mathematical Reasoning", by P. Eccles, Cambridge University Press. ISBN:9780521597180

(I do not require you to buy this book, but you might find it useful.)

**Course description:**

The goal of this course is to learn how to create and write mathematical proofs, and to learn why one might want to do such a thing.
We will introduce and study some important mathematical concepts used in advanced mathematics courses, particularly equivalence relations.

**Assessment:**

There will be homework for most classes, two in class midterm exams and a final exam. Since there will be a lot of writing, explaining and critiquing in class, there will also be a class participation component of the grade.
The relative weighting of these components will be:
Homework: 20%
Class participation: 10%
Midterm exams: 20% each
Final exam: 30%

** Exams:**

Two midterms, and one final, dates and times to be announced.

**The first midterm will be Monday, October 9, in class.**

Here are some worked solutions for Midterm 1.

**The second midterm will be Wednesday, November 15, in class.**

Here are some worked solutions for Midterm 2.

** Daily Homework:**

** For Wednesday, August 30, at the beginning of class.** In class on Monday 8/28, we will work on this worksheet. For the beginning of class on Wednesday, prove Proposition 4. Decide whether or not Conjecture 5 is true. If Conjecture 5 is true, prove it. If it is not true, prove it is not true.

** For Friday, September 1, at the beginning of class.** Prove Propositions 6 and 7 from the above worksheet. If you did not do the Conjecture 5 HW from Wednesday, do this as well.

** For Wednesday, September 6, at the beginning of class.** Prove Propositions 8 and 9 from Worksheet 1.

Here is the second worksheet, which we will beginning working on on Wednesday.

** For Friday, September 8, at the beginning of class.** Prove Propositions 10, 11, 12 and 13 from Worksheet 2.

** For Wednesday, September 13, at the beginning of class.** Do this homework. This homework will be graded (given a score out of 25), and the points will count towards your homework score. [Other homework which is not grade but just read and commented on counts as well, but only for having done it or not.]

Here is the third worksheet, which we will begin working on on either September 13 or 15.

** For Friday, September 15, at the beginning of class.** Prove Propositions 16, 17 and 18 from Worksheet 2.

**For Monday, September 18, at the beginning of class.** Prove Propositions 22 and 23.

**For Wednesday, September 20, at the beginning of class.** In class on Monday, you were given somebody else's homework to grade. Do this grading and come ready to discuss it with that person.

** For Friday, September 22, at the beginning of class**. Prove Lemma 30 and Theorem 31.

**For Monday, September 25, at the beginning of class.** Prove Lemma 30 and Theorem 31. (Again.)

** Here** is a worksheet on sets, relations, and equivalence relations.

**For Monday October 2, at the beginning of class.** Do this homework. It will be graded out of 25 points. Here are some worked solutions for this homework.

Here is the fourth worksheet on number theory.

** For Friday, September 29, at the beginning of class.** Prove statements 1,2 and 3 of Theorem 33, from Number Theory IV.

By popular(?) request, **here** is a proof of Theorem 31.

**For Friday, October 13, at the beginning of class.** Prove statements 6, 7 and 8 from Theorem 33. Also Prove Proposition 34, Theorem 37 and Corollary 38.

Here is the next worksheet, which we will begin working on on Friday 10/13.

** For Monday, October 16, at the beginning of class.** The homework was explained in class. See also here.

**For Wednesday, October 18, at the beginning of class.** Prove Propositions 3 and 4 from the Fields Worksheet.

**For Friday, October 20, at the beginning of class**. Let n be a natural number (at least 2) and let S be the set of equivalence classes of integers mod n. Define addition and multiplication as we did in class (see the equivalence relations worksheet).
Prove that with these operations S satisfies axioms (A1), (A2) and (A3) from the Fields worksheet.

**For Monday, October 30, at the beginning of class**. Do this homework. If will be graded out of 25 points. Here are some worked solutions to this homework.

**For Wednesday, October 25, at the beginning of class.** Prove Propositions 5,6,7 and 8 from the Fields Worksheet.

Here is the next number theory worksheet, which we will start working on on 10/25.

Here is the next number theory worksheet, which we will start working on on 11/3.

**For Wednesday, November 8, at the beginnig of class.** Prove Proposition 54 and Theorem 55.

** For Friday, November 10, at the beginning of class.** Prove the following lemma:

**Lemma.** Let n be a natural number and p be a prime. If n | p then either n =1 or n = p.

Then prove Proposition 54 and Theorem 55 again.

Here is the next worksheet (on induction) which we will start working on on November 17.

**NO CLASS ON WEDNESDAY, November 22.**

**For Monday, November 27, at the beginning of class.** Do Exercise 6 from the worksheet on induction (unless you already turned it in).

**For Wednesday, November 29, at the beginning of class.** Prove Propositions 9 and 10 from the Induction worksheet.

**For Monday, December 4, at the beginning of class.** Prove Proposition 14 from the Induction worksheet.

Here is the next (and probably last) worksheet, which we will work on from Monday, December 4.